This article walks through the steps on how to do this on the new Photobucket experience.
You can select multiple images and videos on your account to perform a bulk action, such downloading or hosting, on the mobile site.
To do this, just tap the Select button in the toolbar, then tap on the images you want to select. You can also try to tap and hold on an image. Once you see a blue checkmark on the top-left corner, you can now tap on multiple images to select them.
To deselect images, you can either tap individual images or tap on the X on the top-left corner to deselect all images.